A bird is flying in the air. To take a turn in the horizontal plane of radius R=10 m with the velocity v=10 m/s at what angle it should bend with the horizontal.

Options

A. `30^(@)`
B. `15^(@)`
C. `60^(@)`
D. `45^(@)`

Explanation

To find the angle at which the bird should bend with the horizontal while flying in a circular path of radius \( R = 10 \, \text{m} \) at a velocity of \( v = 10 \, \text{m/s} \), we can follow these steps:

### Step 1: Understand the Forces Acting on the Bird
When the bird is flying in a circular path, two main forces act on it:
1. The gravitational force (\( mg \)) acting downward.
2. The lift force (\( F \)) acting perpendicular to the wings of the bird.

### Step 2: Resolve the Lift Force into Components
The lift force can be resolved into two components:
- A vertical component: \( F \cos \theta \)
- A horizontal component: \( F \sin \theta \)

Where \( \theta \) is the angle of the lift force with respect to the horizontal.

### Step 3: Set Up the Equations
For the bird to maintain circular motion:
1. The vertical component of the lift must balance the weight of the bird:
\[ F \cos \theta = mg \quad \text{(1)} \]
2. The horizontal component of the lift provides the necessary centripetal force:
\[ F \sin \theta = \frac{mv^2}{R} \quad \text{(2)} \]

### Step 4: Divide the Two Equations
To eliminate \( F \), we can divide equation (2) by equation (1):
\[ \frac{F \sin \theta}{F \cos \theta} = \frac{\frac{mv^2}{R}}{mg} \]
This simplifies to:
\[ \tan \theta = \frac{v^2}{gR} \]

### Step 5: Substitute the Known Values
Given:
- \( v = 10 \, \text{m/s} \)
- \( R = 10 \, \text{m} \)
- \( g = 10 \, \text{m/s}^2 \)

Substituting these values into the equation:
\[ \tan \theta = \frac{(10)^2}{10 \times 10} = \frac{100}{100} = 1 \]

### Step 6: Find the Angle
Now, we can find \( \theta \):
\[ \theta = \tan^{-1}(1) = 45^\circ \]

### Conclusion
The angle at which the bird should bend with the horizontal is \( 45^\circ \).

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